The Hadwiger theorem on convex functions. I

Abstract

A complete classification of all continuous, epi-translation and rotation invariant valuations on the space of super-coercive convex functions on R n is established. The valuations obtained are functional versions of the classical intrinsic volumes. For their definition, singular Hessian valuations are introduced.

The Hadwiger theorem on convex functions, IV: The Klain approach

“…The main aim of this paper is to give a new proof of Theorem 1.1. The proof in [13] followed the basic outline of Hadwiger's original proof. Klain [21] found a different approach to the classical Hadwiger theorem, which we try to adapt to the functional case.…”

“…For valuations on convex functions, the first classification results [9,10,32,33] and the first structural results [4,11,24,25] were recently obtained. In [13], the authors established the following Hadwiger theorem for convex functions. Let…”

“…Theorem 1.1 ( [13]). A functional Z : Conv sc (R n ) → R is a continuous, epi-translation and rotation invariant valuation if and only if there exist functions ζ 0 ∈ D n 0 , .…”

“…In this case, we set ζ(0) := lim s→0 + ζ(s) and consider ζ also as an element of C c ([0, ∞)), the set of continuous functions with compact support on [0, ∞). For 0 ≤ j ≤ n and ζ ∈ D n j , the functionals V n j,ζ : Conv(R n ) → R were introduced in [13], where it was proved that there exists a unique continuous extension to Conv sc (R n ) of the functional defined by…”

“…The authors [13] established the Hadwiger theorem also for valuations on Conv(R n ; R) by using duality with valuations on Conv sc (R n ). For 0 ≤ j ≤ n and ζ ∈ D n j , define V n, * j,ζ as the valuation dual to…”